Question

Find the mass of a thin funnel in the shape of a cone z = √(x² + y²) , 1 ≤ z ≤ 3, if its density function is (x, y, z) = 4 − z.

Answer 1

The mass of a thin funnel-shaped cone with a density function (x, y, z) = 4 - z and dimensions 1 ≤ z ≤ 3 is found by converting to cylindrical coordinates and integrating the density function over the volume of the cone.

Step-by-step explanation:

The task at hand is to find the mass of a thin funnel-shaped cone with a given density function. The equation of the cone is z = √(x² + y²), and it is defined between z = 1 and z = 3. The density function for the cone is described by (x, y, z) = 4 - z.

To calculate the mass, we integrate the density function over the volume of the cone. For a cone, this is a triple integral, which in cylindrical coordinates (r, θ, z) turns into:

To solve the integral step by step:

After solving the integrals, we can sum them up to find the total mass.